⚡ Pluto.jl ⚡

x
 
begin
    anim = @animate for i in range(10, n_steps, step=10)
        i = min(i, n_steps)
        generate_plot(sol, (vals), 1:i)
    end
    gif(anim, "anim_fps15.gif", fps = 10)
end

93.2 s

x
 
begin
    anim2 = @animate for i in range(10, n_steps, step=10)
        i = min(i, n_steps)
        generate_plot(sol2, (vals2), 1:i, τ=-τ)
    end
    gif(anim2, "anim2_fps15.gif", fps = 10)
end

94.8 s

x
 
generate_plot(sol, (vals), :)

251 ms

x
 
generate_plot(sol2, (vals2), :, τ=-τ)

16.8 ms

generate_plot (generic function with 1 method)

x
 
function generate_plot(sol, vals, indices; τ=τ)
    (
        mean_values, below_mean, rolling_mean, 
        mean_rescaled_values, max_rescaled_values, min_rescaled_values,
        median_rescaled_values, q90_rescaled_values, q10_rescaled_values,
        sol_rescaled_matrix, rand_trajectory_indices
    ) = vals
    l = @layout [a; b; c]
    #indices = 1:i
    x = sol.t[indices]
    #p1 = plot(sol, yscale= :log, color=:gray, alpha=0.3, label="")
    p1 = plot(x, mean_values[1,indices], color=:red, label="mean")
    plot!(p1, legend=:outertopright, title="Mean wealth (τ=$τ, n=$n)")
    
    
    p3 = plot(x, below_mean[1,indices], label="below_mean")
    plot!(p3, x, rolling_mean[indices], label="time_avg")
    plot!(p3, legend=:outertopright, title="% below mean value")
    
    
    #p2 = plot(sol.t, sol_rescaled_matrix, color=:gray, alpha=0.3, label="")
    
    p2 = plot(x, mean_rescaled_values[1,indices], color=:red, label="mean")
    plot!(p2, x, max_rescaled_values[1,indices], color=:blue, label="max")
    plot!(p2, x, min_rescaled_values[1,indices], color=:blue, label="min")
    plot!(p2, x, median_rescaled_values[1,indices], color=:blue, label="50%")
    plot!(p2, x, q90_rescaled_values[1,indices], color=:magenta, label="90%")
    plot!(p2, x, q10_rescaled_values[1,indices], color=:magenta, label="10%")
    for i in rand_trajectory_indices
        plot!(p2, x, sol_rescaled_matrix[i,indices], color=:grey, label="")
    end
    
    plot!(p2, legend=:outertopright, title="Rescaled wealth")   
    plot(p1, p2, p3, size=(1000, 600), layout = l)
end

176 μs

$1002$

1×1002 Array{Float64,2}:
 1.0  1.00441  1.00487  1.00757  1.00717  …  -56820.3  -69083.6  -54654.5  -54654.5

1×1002 Array{Float64,2}:
 0.0  0.506  0.527  0.514  0.516  0.526  0.515  …  0.419  0.418  0.414  0.417  0.417

3Float641

$0 . 0$

$0 . 2 5 3$

$0 . 3 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3$

$0 . 3 8 6 7 5$

$0 . 4 1 2 5 9 9 9 9 9 9 9 9 9 9 9 9 7$

$0 . 4 3 1 4 9 9 9 9 9 9 9 9 9 9 9 9 4$

$0 . 4 4 3 4 2 8 5 7 1 4 2 8 5 7 1 4$

$0 . 4 5 2 2 4 9 9 9 9 9 9 9 9 9 9 9 3$

$0 . 4 6 1 1 1 1 1 1 1 1 1 1 1 1 1$

$0 . 4 6 8 1 9 9 9 9 9 9 9 9 9 9 9 9 5$

$0 . 4 7 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5$

$0 . 4 7 9 5 8 3 3 3 3 3 3 3 3 3 3 3$

$0 . 4 8 3 7 6 9 2 3 0 7 6 9 2 3 0 7 3$

$0 . 4 8 6 4 2 8 5 7 1 4 2 8 5 7 1 4$

$0 . 4 8 8 8$

$0 . 4 9 1 3 1 2 5$

$0 . 4 9 3 9 4 1 1 7 6 4 7 0 5 8 8 3$

$0 . 4 9 6 5 5 5 5 5 5 5 5 5 5 5 5 6$

$0 . 4 9 8 7 8 9 4 7 3 6 8 4 2 1 0 5 6$

$0 . 5 0 0 5$

more993

$0 . 5 6 8 9 4 9 6 4 7 5 3 2 7 2 9 3$

994

$0 . 5 6 8 7 9 1 7 5 0 5 0 3 0 1 8 2$

995

$0 . 5 6 8 6 4 1 2 0 6 0 3 0 1 5 0 9$

996

$0 . 5 6 8 4 9 2 9 7 1 8 8 7 5 5 0 3$

997

$0 . 5 6 8 3 3 9 0 1 7 0 5 1 1 5 3 6$

998

$0 . 5 6 8 1 8 9 3 7 8 7 5 7 5 1 5 2$

999

$0 . 5 6 8 0 3 9 0 3 9 0 3 9 0 3 9 2$

1000

$0 . 5 6 7 8 8 5 0 0 0 0 0 0 0 0 0 1$

1001

$0 . 5 6 7 7 3 4 2 6 5 7 3 4 2 6 5 9$

1002

$0 . 5 6 7 5 8 3 8 3 2 3 3 5 3 2 9 5$

1×1002 Array{Float64,2}:
 1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  …  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0

1×1002 Array{Float64,2}:
 1.0  1.12501  1.18447  1.25047  1.33236  …  7586.44  6779.1  8662.02  8662.02

1×1002 Array{Float64,2}:
 1.0  0.828039  0.798682  0.786456  …  -1585.44  -1305.03  -1625.82  -1625.82

1×1002 Array{Float64,2}:
 1.0  0.999422  0.997064  0.996852  0.99739  …  -3.2702  -2.77611  -3.59646  -3.59647

1×1002 Array{Float64,2}:
 1.0  1.05807  1.08189  1.10222  1.11794  …  74.9351  63.5709  82.4535  82.4536

1×1002 Array{Float64,2}:
 1.0  0.943985  0.92129  0.907465  0.8931  …  -89.409  -75.4356  -96.7221  -96.7222

1000×1002 Array{Float64,2}:
 1.0  0.906762  0.916616  0.898629  0.884472  …    1.24771    1.59119    1.59119
 1.0  0.945487  0.858075  0.862511  0.784747     101.542    135.162    135.163
 1.0  0.941019  0.873183  0.851367  0.803476      14.8362    18.6452    18.6452
 1.0  0.946316  0.928038  0.905564  0.883512      14.0832    17.7311    17.7312
 1.0  0.97697   0.921472  0.832453  0.915906      36.9134    46.7759    46.776
 1.0  0.987504  0.889869  0.896703  0.892412  …   15.576     19.3141    19.3141
 1.0  0.999409  1.0398    1.04312   1.10827      -30.0248   -37.1702   -37.1702
 ⋮                                            ⋱               ⋮        
 1.0  0.946462  0.958537  0.961702  0.960493      51.1245    66.7855    66.7855
 1.0  0.981512  0.979709  0.907478  0.91932   …  453.653    599.627    599.627
 1.0  0.936449  0.95218   0.968702  0.945949      -1.0746    -1.40474   -1.40474
 1.0  1.03358   1.04421   1.06579   1.06526       -8.57554  -11.2261   -11.2261
 1.0  1.03495   1.05807   1.01143   0.990378     -18.5972   -22.4368   -22.4368
 1.0  1.00405   1.00977   1.02101   0.992355     -21.5162   -29.1503   -29.1503

11Int641

$467$

$866$

$196$

$841$

$647$

x
 
n_steps, vals  = get_vals(sol)

92.4 ms

$1002$

1×1002 Array{Float64,2}:
 1.0  1.00028  1.00226  1.00322  1.00386  …  7.22464  7.25642  7.28517  7.28517

1×1002 Array{Float64,2}:
 0.0  0.497  0.52  0.523  0.516  0.516  0.531  …  0.565  0.568  0.562  0.572  0.572

3Float641

$0 . 0$

$0 . 2 4 8 5$

$0 . 3 3 8 9 9 9 9 9 9 9 9 9 9 9 9 9 7$

$0 . 3 8 5$

$0 . 4 1 1 2$

$0 . 4 2 8 6 6 6 6 6 6 6 6 6 6 6 6 7$

$0 . 4 4 3 2 8 5 7 1 4 2 8 5 7 1 4 3 4$

$0 . 4 5 3 3 7 5 0 0 0 0 0 0 0 0 0 0 3$

$0 . 4 6 1$

$0 . 4 6 7 6$

$0 . 4 7 3 4 5 4 5 4 5 4 5 4 5 4 5 4 6$

$0 . 4 7 9 6 6 6 6 6 6 6 6 6 6 6 6 7$

$0 . 4 8 3 6 1 5 3 8 4 6 1 5 3 8 4 6$

$0 . 4 8 7$

$0 . 4 9 0 2 6 6 6 6 6 6 6 6 6 6 6 6 3$

$0 . 4 9 3 4 9 9 9 9 9 9 9 9 9 9 9 9 4$

$0 . 4 9 6 4 7 0 5 8 8 2 3 5 2 9 4 1$

$0 . 4 9 8 4 9 9 9 9 9 9 9 9 9 9 9 9 4$

$0 . 5 0 0 7 3 6 8 4 2 1 0 5 2 6 3 1$

$0 . 5 0 2 3$

more993

$0 . 5 6 2 8 1 9 7 3 8 1 6 7 1 6 9 8$

994

$0 . 5 6 2 8 2 0 9 2 5 5 5 3 3 1 9 5$

995

$0 . 5 6 2 8 1 8 0 9 0 4 5 2 2 6 0 8$

996

$0 . 5 6 2 8 2 2 2 8 9 1 5 6 6 2 6$

997

$0 . 5 6 2 8 2 7 4 8 2 4 4 7 3 4 1 6$

998

$0 . 5 6 2 8 2 9 6 5 9 3 1 8 6 3 6 9$

999

$0 . 5 6 2 8 3 4 8 3 4 8 3 4 8 3 4 4$

1000

$0 . 5 6 2 8 3 3 9 9 9 9 9 9 9 9 9 6$

1001

$0 . 5 6 2 8 4 3 1 5 6 8 4 3 1 5 6 4$

1002

$0 . 5 6 2 8 5 2 2 9 5 4 0 9 1 8 1 2$

1×1002 Array{Float64,2}:
 1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  …  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0

1×1002 Array{Float64,2}:
 1.0  1.14538  1.22133  1.28444  1.30007  …  2.33007  2.3128  2.43117  2.43117

1×1002 Array{Float64,2}:
 1.0  0.858851  0.807647  0.779717  …  0.45922  0.465704  0.464225  0.464225

1×1002 Array{Float64,2}:
 1.0  1.00044  0.997208  0.996108  …  0.958374  0.963146  0.963132  0.963132

1×1002 Array{Float64,2}:
 1.0  1.05764  1.08443  1.0969  1.11043  …  1.34074  1.33863  1.33335  1.33335

1×1002 Array{Float64,2}:
 1.0  0.942907  0.922381  0.90541  …  0.713265  0.715867  0.713505  0.713505

1000×1002 Array{Float64,2}:
 1.0  0.997736  0.98027   0.982883  …  0.873308  0.848167  0.791675  0.791676
 1.0  0.950797  0.955326  0.968567     0.722607  0.723182  0.713573  0.713573
 1.0  1.01378   0.990531  0.963829     0.729599  0.733584  0.712423  0.712424
 1.0  0.95753   0.936931  0.961542     0.737401  0.713991  0.683079  0.683079
 1.0  0.918856  0.965992  0.963822     1.51141   1.57924   1.61247   1.61247
 1.0  0.955954  0.996973  1.00992   …  1.09496   1.09423   1.07979   1.07979
 1.0  0.964941  0.973986  0.985951     0.768035  0.758382  0.770829  0.770829
 ⋮                                  ⋱                      ⋮         
 1.0  1.02924   0.956967  0.955749     1.14021   1.17685   1.18437   1.18437
 1.0  1.03407   1.045     1.05124   …  0.996936  0.963976  0.999022  0.999021
 1.0  1.08054   1.11138   1.13716      0.657496  0.652194  0.653937  0.653936
 1.0  1.03727   1.01601   1.03225      1.3156    1.30934   1.31979   1.31979
 1.0  1.04184   1.04809   1.03317      0.99756   1.02491   1.11198   1.11198
 1.0  1.05685   1.09597   1.11517      0.567786  0.605907  0.581047  0.581047

11Int641

$315$

$487$

$323$

$744$

$881$

x
 
n_steps2, vals2 = get_vals(sol2)

88.8 ms

get_vals (generic function with 1 method)

x
 
function get_vals(sol)
    sol_matrix = hcat(sol.u...)
    mean_values = mean(sol_matrix, dims=(1))
    n_steps = length(sol.t)
    below_mean = mean(sol_matrix .< mean_values, dims=1)
    rolling_mean = cumsum(below_mean, dims=(2))[1,:] ./collect(range(1.0, n_steps, step=1.0))
    sol_rescaled = deepcopy(sol)
    sol_rescaled_matrix = sol_matrix ./ mean_values
    #sol_rescaled.u = [r[:] for r in eachrow(sol_rescaled_matrix)]
    mean_rescaled_values = mean(sol_rescaled_matrix, dims=(1))
    max_rescaled_values = maximum(sol_rescaled_matrix, dims=(1))
    min_rescaled_values = minimum(sol_rescaled_matrix, dims=(1))
    median_rescaled_values = median(sol_rescaled_matrix, dims=(1))
    q90_rescaled_values = hcat([quantile(r[:], 0.9) for r in eachcol(sol_rescaled_matrix)]...)
    q10_rescaled_values = hcat([quantile(r[:], 0.1) for r in eachcol(sol_rescaled_matrix)]...)
    rand_trajectory_indices = rand(1:n_steps, 5)
    vals = (
        mean_values, below_mean, rolling_mean, 
        mean_rescaled_values, max_rescaled_values, min_rescaled_values,
        median_rescaled_values, q90_rescaled_values, q10_rescaled_values,
        sol_rescaled_matrix, rand_trajectory_indices
    )
    return n_steps, vals
end

119 μs

[36mSDEProblem[0m with uType [36mArray{Float64,1}[0m and tType [36mFloat64[0m. In-place: [36mfalse[0m
timespan: (0.0, 100.0)
u0: [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

x
 
begin
    μ=0.021
    σ=0.14
    τ=-0.15
    n=1000
    tspan=(0.0, 100.0)
    prob = get_diff_eq(μ=μ, σ=σ, τ=τ, n=n, tspan=tspan)
end

8.9 s

prob2

[36mSDEProblem[0m with uType [36mArray{Float64,1}[0m and tType [36mFloat64[0m. In-place: [36mfalse[0m
timespan: (0.0, 100.0)
u0: [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

x
 
prob2 = get_diff_eq(μ=μ, σ=σ, τ=-τ, n=n, tspan=tspan)

194 μs

xxxxxxxxxx
 
using DifferentialEquations

47.7 s

Plots.PyPlotBackend

xxxxxxxxxx
 
using Plots; pyplot() #plotly() # Using the Plotly backend

23.4 s

xxxxxxxxxx
 
using Statistics

32.4 ms

Float64

xxxxxxxxxx
 
typeof(1/2^(4))

6.9 μs

get_diff_eq (generic function with 1 method)

xxxxxxxxxx
 
function get_diff_eq(;
        μ::Number=1.0, 
        σ::Number=1.0, 
        τ::Number=0.0, 
        n::Number=2, 
        tspan::Tuple{Number,Number}=(0.0, 100.0), 
        #u₀::Array{Number}=nothing
    )
    u₀ = ones(n)
    f(u,p,t) = μ*u - τ*(u .- mean(u))
    g(u,p,t) = σ*u
    prob = SDEProblem(f,g,u₀,tspan)
    return prob
end

213 μs

retcode: Success
Interpolation: 1st order linear
t: 1002-element Array{Float64,1}:
   0.0
   0.1
   0.2
   0.30000000000000004
   0.4
   0.5
   0.6
   ⋮
  99.59999999999862
  99.69999999999861
  99.7999999999986
  99.8999999999986
  99.9999999999986
 100.0
u: 1002-element Array{Array{Float64,1},1}:
 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
 [0.998019015752889, 0.9510658414926022, 1.0140694777315076, 0.9578012096701445, 0.919115750512006, 0.9562252003478064, 0.9652141479108769, 0.9706803007418675, 1.0693271786638994, 1.005725074125227  …  1.0260034142387402, 0.9774434436720234, 1.0422001114652972, 1.0055480480456966, 1.0295297866000548, 1.0343606289661411, 1.0808454876960563, 1.037564871012976, 1.0421317418173428, 1.0571462217470442]
 [0.9824844006466523, 0.9574841946060301, 0.9927687303308445, 0.9390475395041075, 0.9681739420261597, 0.9992250635127821, 0.9761864988499609, 0.9749777440755799, 1.1185920911819842, 0.9996010984096022  …  1.0128148913950608, 0.9643601311596616, 1.0871619763929496, 1.0071108110479834, 0.9591293404348149, 1.0473567576765512, 1.1138866594668928, 1.0183030903754369, 1.0504537398543374, 1.098447284362764]
 [0.9860463848081689, 0.9716847501087723, 0.9669308623772892, 0.9646365640051681, 0.9669246811117161, 1.0131717110854639, 0.9891246079945577, 1.0543268531059922, 1.1732885626077654, 1.0152492187419375  …  1.0129504149464972, 0.9727387036196269, 1.0630960079494067, 1.0176841188785029, 0.9588252695398269, 1.0546222263836145, 1.140818110044002, 1.0355764384147492, 1.036495271292124, 1.1187574179226343]
 [1.0085711020573354, 0.9688124576973165, 0.9731906203406078, 0.8875361566189837, 0.9337511221653139, 0.9917939300864601, 0.9711477516182847, 1.1034427215447837, 1.2381125012644685, 0.9718812833537678  …  1.053858781600805, 1.0088169286450392, 1.0233671401856204, 1.0105631872778331, 0.986423029088711, 1.106337284755494, 1.0571880222513896, 1.0279559033203622, 1.062273669287912, 1.0744752535914042]
 [0.9985448018002562, 0.9890005186576608, 0.9814689545753578, 0.8657904412760288, 0.9652600449580027, 0.9654300719754308, 0.9410039754317325, 1.0909291826622525, 1.2736813425671867, 1.0005982339652475  …  1.0331588937786556, 0.9534195061940692, 1.0505014177103604, 1.0226781364606887, 0.9381766550748648, 1.0908547230437289, 0.9666615732892213, 1.0158855551936894, 1.1867798328282506, 1.141474031208177]
 [0.9979176373268588, 0.9677737210159575, 0.9731578453695062, 0.8386383469588713, 0.9092004841646534, 0.9653225608363216, 0.9004746679142716, 1.116162018784377, 1.226907049852087, 0.9847765727788872  …  0.994266256117669, 0.9831930560695477, 1.0838941959280866, 0.985100017865773, 0.9028551378884836, 1.1072539076277403, 0.9697196118730311, 1.0838009688192383, 1.1887551374310195, 1.1301885781373155]
 ⋮
 [6.007617830026853, 5.518750059680582, 5.208741916089139, 4.9526517022883745, 10.717076559978665, 7.178555965772261, 5.78486641092313, 5.846649048401186, 7.127984106953892, 7.74830558098805  …  4.628301277046172, 4.9016179248704, 7.077297282476337, 6.589564669243558, 8.53420124280035, 6.671835660288932, 4.94988196662736, 8.904778362042173, 7.323413010417348, 4.287439123817096]
 [6.173959160967872, 5.364251369804051, 5.247376240590091, 5.099542032946705, 11.094601933625011, 7.764965030854282, 5.529084425099932, 6.040577700122486, 7.250356506418934, 7.811905893952469  …  4.863641853013579, 4.774401730694823, 6.937836279355249, 6.853764932166131, 7.995241148298569, 7.0448134838736935, 4.8801853329138964, 9.157465845073196, 7.06475436088209, 4.369006478703452]
 [6.30933043468796, 5.220569794041702, 5.2710862516930135, 5.327456598058746, 10.919420068282582, 7.910723182676571, 5.548771339988939, 6.339042836253449, 6.720111885860936, 7.5351288104820675  …  4.594848297650571, 4.691698618421832, 6.940738679965065, 7.0549749306087195, 8.237599513823463, 7.202496228653328, 4.750167242943078, 9.504738438020553, 7.207004525988839, 4.102050277005823]
 [6.154656444749044, 5.247713695808727, 5.323191087075731, 5.181014277902353, 11.459615652070859, 7.940175370971785, 5.503139422508213, 6.797889948885585, 6.8397785043376915, 7.659744915814635  …  4.68146931874797, 4.7683275457256, 7.416770606550524, 6.847848253863965, 8.539712640187402, 6.995013967970305, 4.732595626015025, 9.50109794046614, 7.437169824974048, 4.396714226900677]
 [5.767490375177858, 5.198504465040368, 5.190126802966306, 4.976346772957234, 11.74710704521954, 7.866463333387993, 5.615621220436422, 6.6074258655017175, 6.516238283967131, 8.076917370797762  …  4.9988256876993535, 4.96913448861327, 7.18312593044942, 6.9249006705389204, 8.628344057868167, 7.278043384656616, 4.764039686985904, 9.61492984870188, 8.101000602930913, 4.233026003385014]
 [5.7674920298146075, 5.198504302704035, 5.19012800986175, 4.976346616834273, 11.747106687349568, 7.86646181150715, 5.615620287656042, 6.607425348017516, 6.516237627450753, 8.076914895411957  …  4.998824863734882, 4.969134897876926, 7.183125910726612, 6.924900610903935, 8.628345711685018, 7.278041321661102, 4.764038905983389, 9.614930051574952, 8.10100132320552, 4.233025305700403]

x
 
begin
    dt=0.1
    sol = solve(prob,EM(),dt=dt)
    sol2 = solve(prob2,EM(),dt=dt)
end

148 ms

Array{Float64,2}

xxxxxxxxxx
 
typeof(hcat(sol.u...))

5.9 ms

Int641

$171$

$512$

$570$

$713$

$245$

$847$

$681$

$674$

$573$

$105$

xxxxxxxxxx
 
rand(1:n_steps, 10)

5.3 μs

xxxxxxxxxx
 
html"""<style>
main {
    margin: auto;
    max-width: 1400px;
}
"""

3.7 ms